So I stumbled over following definition of the Cantor Function $f_n$:
Let $$g_n(x) := \left(\frac{3}{2}\right)^n \unicode{x1D7D9}_{\{x \in C_n\}}(x)$$
Where $C_n$ is the n-th Cantor Set.
Then $f_n$ is defined as: $$f_n(x) = \int_{0}^{x}g_n(t)dt$$
This is a beautiful definition (if you know how to integrate this), since it follows that $f_n$ is continous and increasing.
But how do I integrate this?
I know that $\int_{-\infty}^{\infty}\unicode{x1D7D9}_{\{x \in A\}}(x)dx = \int_{A}dx$. But that does not help me since we are integrating to $x$ and not $1$.
EDIT:
I believe the following is correct:
$$f_n(x) = \int_{0}^{x}g_n(t)dt = \left(\frac{3}{2}\right)^n length(C_n\backslash(x, 1])$$
where $(1, 1] = \emptyset$.
The cantor function is the limit of this seqence of functions $f_n$. So the nice properties you cite don't follow from this definition without a little extra argument. The "n-th Cantor set" you refer to is the n-th stage in the construction of the cantor ternary set. The integral is just the total length of the support of the nth stage up to the point $x$. In other words this is a pretty tame function and integration is straightforward.
For a picture see the "iterative construction" on the Wikipedia page. $f_n$ is the nth stage of this construction. https://en.m.wikipedia.org/wiki/Cantor_function